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Knots Unravelled: From String to Mathematics

Meike Akveld and Andrew Jobbings
Publication Date: 
Number of Pages: 
[Reviewed by
Scott Taylor
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In his classic Knot Book, Colin Adams relays the story of a hapless undergraduate who inquires of his calculus professor, “What kind of math do you study”. When the professor replies, “Knot Theory,” the student responds, “Oh good! I don’t like theory either!” The misunderstanding aside, the student’s response points to one of the delights of knot theory: one can learn and understand a fair amount of this subject without needing a lot of mathematical knowledge or technique. Indeed, knot theory might be the perfect way of enticing a mathematically bored middle or high school student into deeper and more interesting waters.

Knots Unravelled is a wonderful gift for such a student. It is slim, easy to read, and filled with delightful mathematical ideas. After beginning with a history of knots and knot theory, each chapter addresses some aspect of the mathematical theory of knots. The topics range from knot diagrams and the Reidemeister moves, to crossing number, unknotting number, 3-colorability, and the Jones polynomial. Only the last chapter, on the Jones polynomial, assumes any high-school algebra. Between the chapters are interludes which explore specific examples of knots. These include the Celtic knots and the more prosaic but tractable torus knots.

The organization and writing of the book are superb. I read it in an hour on a plane flight. The discussion is appropriately informal and is interspersed with exercises called “tasks.” Many of the tasks are straightforward but non-trivial. A dedicated high school student could enjoyably pass quite a bit of time drawing pictures or playing with string in an effort to solve them. Solutions to the tasks are in the back, so even if the student is not dedicated, they can still learn how to solve the problems.

For readers who want more mathematical rigor and depth, the aforementioned book by Adams is a great place to start. From there, the reader can progress to any number of other undergraduate and graduate texts.


Colin Adams, The Knot Book, American Mathematical Society, 2004

Scott Taylor is a knot theorist at Colby College who is just occasionally “not a theorist.”

  1. Introduction
    1. Knots everywhere
    2. Knots in rope
    3. Knot science
    4. History

Interlude Knots in paper

  1. Working with diagrams
    1. Describing knots
    2. Mathematical knots
    3. Projections and knot diagrams
    4. Knotted or not? The same or different?
    5. Reidemeister moves

Interlude Celtic knots

  1. Counting crossings
    1. Telling knots apart
    2. The crossing number
    3. Which crossing numbers are possible?
    4. Does the crossing number classify knots?
    5. Crossing number 5
    6. Classifying knots

Interlude Tie knots

  1. New knots from old
    1. Mirror images
    2. Combining knots
    3. Changing crossings

Interlude The figure of eight

  1. Using colours
    1. Knot invariants
    2. Three-colourability

Interlude Hunter's bend

  1. Links
    1. What is a link?
    2. The Borromean rings
    3. Components
    4. The linking number
    5. Three-colourability

Interlude Torus knots

  1. Knot polynomials
    1. The bracket polynomial
    2. The writhe
    3. The X-polynomial
    4. The Jones polynomial

Postlude A special trefoil



Table of knots and links